Nova explosions occur on the white dwarf (WD) component of a
Cataclysmic Variable stellar system which is accreting
matter lost by a companion. A type Ia supernova is thought
to result when the WD, with the same configuration, grows in
mass to the Chandrasekhar Limit. Here, we present the first
self-consistent calculations of accretion onto massive WDs
which demonstrate that this evolution can occur. In this
study, we have investigated the accretion of Solar matter at
high mass accretion rates onto hot, luminous, WDs
(L>30L\odot). We use the 1D, hydrodynamic, computer
code described in Starrfield et al. (APJS, 127, 485, 2000)
plus a nuclear reaction network found at {\tt
http://flash.uchicago.edu/fxt/code-pages/net-pphotcno.shtml}.
In contrast to our nova simulations, where material is
ejected at high velocities \it only when the WD has a low
luminosity and the rate of mass transfer onto the WD is <
10-9M\odot yr-1, these simulations do not eject
material and the WD grows in mass. Our simulations also show
that a hydrogen thermonuclear runaway does not occur since
hydrogen fuses immediately to helium in the surface layers.
As the helium ash grows in mass, it gradually fuses both to
carbon and more massive nuclei depending on the WD mass and
mass accretion rate. Moreover, the surface conditions of our
simulations agree with observations of the Super Soft X-ray
binary sources. Our results, therefore, confirm previous
suggestions that some of the Super Soft X-ray binaries (CAL
83 and CAL 87, for example) may be the progenitors of SN Ia
explosions. S. Starrfield acknowledges support from NSF and
NASA grants to ASU. FXT is supported by DOE under Grant
No.~B341495 to the Center for Astrophysical Thermonuclear
Flashes at the University of Chicago. EMS is supported by
NASA ADP grant NAG5-11182.